Option Pricing - Incorrect price outcome for Out of the Money (OTM) calls

Question

I have the options data for a stock -

structure(list(Curr_Date = structure(c(18904L, 18904L, 18904L, 
18904L, 18904L, 18904L, 18904L, 18904L, 18904L), class = c("IDate", 
"Date")), ticker = c("GOLD", "GOLD", "GOLD", "GOLD", "GOLD", 
"GOLD", "GOLD", "GOLD", "GOLD"), ExpDate = structure(c(18915L, 
18915L, 19013L, 19013L, 19013L, 19377L, 19377L, 19377L, 19377L
), class = c("IDate", "Date")), Strike = c(18, 30, 10, 30, 40, 
10, 18, 30, 40), Option_Type = c("calls", "calls", "calls", "calls", 
"calls", "calls", "calls", "calls", "calls"), OI = c(3570L, 341L, 
723L, 68772L, 26302L, 1731L, 15662L, 37274L, 13215L), Vol = c(1L, 
1L, 5L, 40L, 1L, 1L, 2L, 4L, 5L), ask = c(0.56, 0.01, 8.6, 0.07, 
0.04, 10, 2.8, 0.51, 0.21), bid = c(0.54, 0, 8.2, 0.06, 0.03, 
8.05, 2.58, 0.48, 0.2), StockPrice = c(18.23, 18.23, 18.23, 18.23, 
18.23, 18.23, 18.23, 18.23, 18.23), days2exp = c(0.0301369863013699, 
0.0301369863013699, 0.298630136986301, 0.298630136986301, 0.298630136986301, 
1.2958904109589, 1.2958904109589, 1.2958904109589, 1.2958904109589
), calculated_price = c(0.53, 0, 8.27, 0, 0, 8.5, 2.87, 0.36, 
0.06)), row.names = c(NA, -9L), class = c("data.table", "data.frame"
), .internal.selfref = <pointer: 0x562667c9eac0>)

which looks like -

    Curr_Date ticker    ExpDate Strike Option_Type    OI Vol   ask  bid StockPrice   days2exp calculated_price
1: 2021-10-04   GOLD 2021-10-15     18       calls  3570   1  0.56 0.54      18.23 0.03013699             0.53
2: 2021-10-04   GOLD 2021-10-15     30       calls   341   1  0.01 0.00      18.23 0.03013699             0.00
3: 2021-10-04   GOLD 2022-01-21     10       calls   723   5  8.60 8.20      18.23 0.29863014             8.27
4: 2021-10-04   GOLD 2022-01-21     30       calls 68772  40  0.07 0.06      18.23 0.29863014             0.00
5: 2021-10-04   GOLD 2022-01-21     40       calls 26302   1  0.04 0.03      18.23 0.29863014             0.00
6: 2021-10-04   GOLD 2023-01-20     10       calls  1731   1 10.00 8.05      18.23 1.29589041             8.50
7: 2021-10-04   GOLD 2023-01-20     18       calls 15662   2  2.80 2.58      18.23 1.29589041             2.87
8: 2021-10-04   GOLD 2023-01-20     30       calls 37274   4  0.51 0.48      18.23 1.29589041             0.36
9: 2021-10-04   GOLD 2023-01-20     40       calls 13215   5  0.21 0.20      18.23 1.29589041             0.06

As it can be seen in column calculated_price, the OTM (out of the money) calls are hugely underpriced compared to bid or ask price. I have used the below formulas to calculate the expected price of an option.

r = 0.0148 # Risk free rate
v = 0.3188 # Historic volatility of 252 trading days
b = 0.02 # TTM yeild
dt[, d1 := ((log(StockPrice/Strike)) + (r - b + (v^2)/2) * days2exp)/(v * (sqrt(days2exp)))]
dt[, d2 := d1 - v * (sqrt(days2exp))]
dt[, calculated_price := round(StockPrice * pnorm(d1) - Strike*exp(-r * days2exp)*pnorm(d2), 2)]

Can someone point me out what is wrong with these formulas and how to correct the mispricing?

The expected result is - The values in column calculated_price should be between the market price in bid and ask columns.

Thanks!

Answer 1

Your end goal of obtaining "fair" theoretical option prices will unfortunately require a lot of effort if you want to get this done properly on your own.

Here are a few reasons why:

• All Nasdaq marketplace stock options are American-style
• IVOL exhibits a skew
• You need to have reliable interest rates
• Dividend assumptions will influence your pricing for longer tenors a fair bit

The forward price is not directly quoted for listed options markets. Futures may be quoted but maturities frequently do not coincide with (all) option maturities. Interpolation is not trivial because future dividends (even times of payments) are usually unknown. Therefore, most practitioners use vanilla equity options to back out (implied) dividends.

The problem hereby is that put-call parity for American-Style options does not hold. Even for European-Style options, there are frequently issues with different trading times and erratic option prices etc. Since dividend payments are discrete in nature, you require a dividend schedule with dates and amounts to derive an implied dividend curve. This curve is usually noisier than one would hope for and commonly smoothed (via Kalman filter of the like).

For American options, one needs to first compute the European equivalent via a process called de-Americanization. For tenors that are not liquid, extrapolation may also be needed.

For interest rates, it is customary to use stripped interest rate curves. These used to be Libor 3m based but are now mainly SOFR curves, which are fundamentally built in a similar way.

Lastly, your main task will be building a reliable vol surface. SVI is frequently used. Voladynamics provides some interesting ideas and examples.

Your formula seems to exclude $e^{-q}$. As Wikipedia correctly states, it should be $${\displaystyle Se^{-q\tau }\Phi (d_{1})-e^{-r\tau }K\Phi (d_{2})\,}$$ This formula also requires rates and dividends to be continuous.

This link shows data sources (also IVOL). I am not familiar with any of these to be honest, but potentially they will help (be reliable). Many vendors like Bloomberg also offer, at a cost, Vol surfaces as part of their standard product offering.